1. Introduction

Since the first paper of E. Ott, C. Grebogi and J.A. Yorke [1], who have shown that given a chaotic system one can convert it into the system exhibiting regular periodic motion, the interest in the field called control of chaos is not weakening. The problem of stabilizing the unstable dynamics concerns many nonlinear systems, among them lasers. Chaotic dynamics and the possibility of its control has been investigated, both theoretically and experimentally, in few lasers; e.g. CO₂ laser [2–5], solid-state Nd:YAG laser [6], ¹⁵NH₃ laser [7], semiconductor lasers [8–10], and recently the Kerr lens mode locked Ti:Sapphire laser [11]. As to our knowledge the solid state transition metal ion lasers have not been widely studied from this point of view. Nowadays the alexandrite laser (chromium doped chrysoberyl crystal) has gained high interest due to its wide applicability in medicine, cosmetics and as a main part of a lidar in investigations of the atmospheric pollution. This is the reason why the control of its stability and the type of its output appears to be a crucial problem.

In our recent papers we have shown the peculiar dynamics of the alexandrite laser in its vibronic mode of operation [12–15]. Depending on frequency and power of the pumping laser it exhibits completely different types of output; e.g., CW operation, self-pulsations, homoclinic dynamics and, in the region of two latter ones, the chaotic dynamics. We have provided the theoretical explanation of the observed phenomenon assuming that the nonequilibrium phonons of the host crystal actively participate in the laser action [16]. The laser model presented by us [12–16] appeared to describe well the experimental results obtained by us [12–15] and other authors [17]. It has been shown experimentally [12–15, 17] that, when pumped by a short wavelength laser (<600nm), both by a cw-laser [12–15] and by a pulsed laser [17], the alexandrite laser operates as a train of pulses, whereas long wavelength pump (>600nm) [12–15, 17] yields the stable cw output. Thus, the pump wavelength can be a control parameter. Moreover we have demonstrated that, for short wavelength pump and certain values of other control parameters such as pump intensity or cavity losses [13, 14], the alexandrite laser output exhibits homoclinic or chaotic character. We have shown that in this range of parameters the vibronic laser dynamics can be well described in terms of the homoclinic orbits and Shil’nikov chaos [13, 14].

In this paper we present the theoretical study on the possibility of control of the alexandrite laser chaotic dynamics. Our numerical results prove that using the time delayed incoherent optical feedback, according to the Pyragas method [18], assures the expected result. Moreover, we show that in the system of two nearly identical vibronic lasers, where one is driven by the output of the other the ideal synchronization can be achieved. Thus, the system might be used for secure message transmission.

2. Theory

In the theoretical model proposed by us [12–16] the dynamics of the nonequilibrium host lattice phonons plays the crucial role. The peculiarity of the vibronic alexandrite laser lies in the fact that the upper lasing level has the energy higher than the storage level [19]. Thus, the pumping process takes place with participation of phonons of energy fitting to the energy gap between the storage level and the upper lasing level. Moreover these are not the thermal phonons [16], but the coherent nonequilibrium phonons obtained in the process of nonradiative depopulation of the pumped level to the storage level. Although the alexandrite laser is a multimode laser, we have proved [12, 15, 16] that the one mode model is sufficient to visualize the variety of laser outputs for different conditions. The laser dynamics is connected with the competition between phonons and photons produced in the system and not with its multimode character. The central mode of the laser is much stronger than the other ones [16]. We have shown [15, 16] that the dynamics of the laser, observed by us experimentally [12–14], can be well described by the rate equations for the following variables: W is the population inversion between the lasing levels, n₃ is the population of the storage level, I is the number of laser photons and N is the number of nonequilibrium phonons active in the pumping process. In order to introduce the control of chaotic dynamics we apply the Pyragas method [18] of the time delayed feedback control. It is a convenient method in continuous dynamical systems. Thus, we introduce the optical intensity feedback in the set of equations derived by us previously for one laser mode [12, 15, 16]:

where n ₀ is the Cr-ion concentration density, κ = cT /2L (c is the light velocity, T is the transmission of the mirror, L is the cavity length) denotes the cavity losses, T_p is the phonon decay rate, C is the electron-field coupling coefficient, B is the electron-phonon coupling coefficient, I ₀ is the number of pumping photons, and the other coefficients are N -dependent:

The parameter _f is the result of the energy conservation [12, 15, 16]. It has the sense of the mean order of nonradiative multiphonon transition from the pumped level to the storage level. It is a very important parameter, because it includes the information about the wavelength of the pumping laser [12, 16]. The higher pump wavelength the larger value of the parameter f.

The delayed term in Eq. (1)c) is characterized by parameters: ε as the control weight parameter, ∣ε∣ < 1, and τ as a time delay.

3. Numerical results

According to the Pyragas method [18] stabilization of a particular unstable periodic orbit can be achieved if τ = T_i, where T_i is the period of the i-th orbit. In our system we are dealing with the periodicity connected with repetition of pulses when the laser output has the form of the regular pulse train [12–14], whereas the shape and the height of pulses depends on the pump wavelength [14]. The chaotic output, obtained in the period doubling route, has the form of irregular pulsations [13, 14]. Thus it consists of the set of unstable periodic pulse trains (the unstable periodic orbits in our laser). Numerical calculations have been performed for different values of the delay time τ , putting it equal to T or T/n, T-being the period of the unstable orbit, n-denotes the natural number. The results of the calculations are shown in Fig. 1 for ε > 0. The case ε > 0 describes the experiment of optical delayed feedback, in which the delayed output laser signal is injected into the laser cavity. It can be seen in Fig. 1 that we get evident control of the chaotic behavior of the system also in the case when the period of the i-th orbit is the high multiplicity of the delay time. This information is important from the point of view of the experimental realization of the predicted here theoretical picture, which yields shorter delay path.

 

Fig. 1. Laser output vs. time in the regime of self-pulsations; a) chaotic dynamics in the absence of the control term ε = 0; b) and c) stabilization of the orbit of the period T = 1.7μs at ε = 5 10^-5 and the delay time: (b) τ = T or (c) τ = T/40; d) and e) stabilization of the orbit of the period T = 2.2μs at ε = 5 1031 and the delay time: (d) τ = T or (e) τ = T/5.

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4. Synchronization of two alexandrite lasers

Herewith we consider two identical lasers in a driving system- response system configuration [20]. The delayed output of the first laser is injected into the cavity of the second one, what is described by Eqs. (2).

where the parameters Γ_i3,Γ_iW,K _3iW,K _3i0,K _wi3,K _Wi0,K_NiW,K _Ni3,K _Ni0 are defined in Eq. (1), whereas N is substituted by Ni, for i = 1 or 2 respectively.

The numerical solutions of Eqs. (2) are shown in Fig. 2 for ε = 0 and ε = 0.0006 . It can be seen in Fig. 2a,b,c that in the absence of coupling, ε = 0, the outputs of two lasers with identical parameters but different initial conditions, remain uncorrelated. On the other hand when the small amount, ε = 0.0006, of the I₁ output (called the transmitter signal) is injected into the cavity of the I₂ laser (called the receiver laser) we observe high synchronization of both signals, what becomes apparent if one compares Fig. 2a and Fig. 2d. Thus the system of two alexandrite lasers may be used for secure transmission of the encoded message. We follow the idea realized experimentally in the system of two Nd: YVO₄ microchip lasers by U. A. Uchida et. al [21]. The chaotic output of the driving laser, treated as transmitter, is modulated by the encoded signal of the amplitude much smaller than the amplitude of the carrying signal. In our case we put the coupling parameter in Eq. (2g) to be time dependent: $\epsilon \left(t\right)=\{\begin{array}{c}\multicolumn{1}{c}{{\epsilon }_0+\Delta \epsilon \mathrm{for}{t}_i<t<t_i+\mathrm{\Delta t}}\\ \multicolumn{1}{c}{{\epsilon }_0-\Delta \epsilon \mathrm{for}{t}_i+\Delta t<t<t_{i+1}}\end{array}.$ Thus, the slave laser, called the receiver, is synchronized to the modulated signal. The encoded message can be read as a difference between the receiver and the unmodulated transmitter signals. The results of our calculations are shown in Fig. 3.

5. Summary

Herewith we have presented the possibility of control of chaos in the vibronic laser by use of the delayed optical feedback. We have also shown that two vibronic lasers in chaotic regime of operation can be synchronized and thus applied for secure communication.

 

Fig. 2. Numerical solutions of Eqs. (2). Laser outputs of two unsynchronized lasers in chaotic regime: a), b), c) and for two synchronized lasers: a), d), e).

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Fig. 3. Numerical solutions of Eqs. (2) with time dependent coupling parameter ε(t) : a) signal from the transmitter laser, b) signal from the transmitter laser modulated by encoded message for ε ₀ = 1, ∆ε = 0.008, c) signal from the receiver laser synchronized to the signal in (b), d) difference signal between (c) and (a).

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